Regressions and Econometric Results

Autocorrelated error terms represent a violation of one of the assumption of the classical linear regression models. Indeed one of the major assumptions of CLRM dictates that error terms should be serially uuncorrelated, i.e. E (utut-j) = 0 for j = 1,2…. Where u is the error term and subscript t represents time. The effect of autocorrelation on inference is that it makes the standard error biased, and hence of course affects the t-statistics and F-statistics. Most macroeconomic time series tend to exhibit positive autocorrelation. And this tends to lead to a downward bias in the standard error. Obviously, the t-statistic will be inflated, hence leading to the possibility of rejecting a null hypothesis when it is true. In the following paper, we provide an example of a model that exhibits serially autocorrelated error terms. We test for autocorrelation and attempt to correct for it. A Cochrane-Orcutt procedure with an autoregressive error process of order 2 was found to be the better model.
The objective of this research is to model the rate of change (proxied by the natural logarithm) of consumption in the UK for the period 1948 to 1997. A simple (specific to general) modelling approach was used where by three variables were initially used to model the logarithm of the consumption. The three explanatory variables were logarithm of income, inflation rate and the logarithm of the interest rate. Ordinary Least Squares (OLS) is used initially. The original equation to be estimated is:
The results invoke the problem of positive serial autocorrelation. Indeed, the Durbin Watson d -statistic implies that the null hypothesis of no serial correlation is rejected in favour of positive serial correlation (ρ>0). The calculated d (0.395997) turns out to be less than the lower d with numerator 49 and denominator 4 (1.38). Hence we cannot adequately perform inference based on the t-statistics and the F-statistics since the standard error term of the coefficients are understated. At a glance the t-statistics and the f-statistics offer reasonably good results, the possibility of making Type 1 errors exist. As a further informal check, the graphical plots of the residuals are plotted.
From the graph, it is easy to see that positive values of the residuals are followed by positive values of the residuals. The residuals appear far from being randomly distributed.
Since the degree of autocorrelation (ρ) is unknown, we need to model the autocorrelation by using the Cochrane-Orcutt (1949) procedure or the Prais and Winsten (1954) estimator. The Cochrane Orcutt estimation procedure is used, first with an autoregressive error term. In this case, one additional observation is lost and this is not desirable since we are dealing with a small sample. The new equation estimated is
The reason for inserting a lag error term is that it takes into account dependence from past error terms. Since the problem with autocorrelation is that there is dependence from lagged values of the error terms, modelling this dependence through the maximum likelihood procedure (from which the Cochrane Orcutt procedure is based) will lead to a fall in the level of autocorrelation.
The rationale for using the Cochrane -Orcutt was to eradicate the autocorrelation in the error term. This objective was achieved partly. It is worth noting from the Durbin -Watson statistic that this value is very close to the brink of rejection of the null hypothesis. Hence, again, the conventional t-statistics and the F-statistics are irrelevant in decision making. It is worth noting that the coefficient of the lagged error term (ρ) turns out to be 0.80, positive and significant. There appears to be weak persistence in the residual term. Intuitively, if we add a lag of second order in the model, the positive autocorrelation may disappear. Figure 2 shows the fitted values and the plot of the residuals.
Figure 2 shows that the plot of the residual terms has greatly improved in the case of the Cochrane-Orcutt procedure as opposed to the case where the ordinary least squares is used. The residual plot shows more randomness and independence. However, we still find that the error terms tend to be beyond the N (0,1) boundaries at times. We expect that as we increase the lag orders of the residuals, the positive autocorrelation shall disappear such that inference may be done on the results. We can therefore estimate the Cochrane-Orcutt procedure by assuming an autoregressive process of order 2 of the residuals. The AR (2) residuals equation is given by
In this case we find that autocorrelation has disappeared completely. The Durbin Watson d statistic of 2 is ideal and refutes completely any claims of the positive autocorrelation found before. Hence in this set up, the model with 2 lagged error terms explains adequately the logarithm in consumption. This conforms to theory in macroeconomics where shocks are known to have lagged impacts on macroeconomic variables.
Figure 3, which is the residual plot of the error terms for the Cochrane Orcutt procedure, shows a considerable improvement. The error terms appear to be independently identically distributed.
Another way to tackle the autocorrelation in the regression is to use the augmented distributed lag technique (ADL). In time series, this technique is often used to remove autocorrelation. It simply implies adding lags of the dependent variable as regressors in the regression equation. Including the lagged dependent variable will actually imply that that the dependence that was previously present is being modelled.
The regression does not appear better. We cannot effectively conclude whether autocorrelation has been cured or not. Since, the Durbin-Watson d test cannot be applied when there is a lagged dependent variable in the regression. In that case, a portmanteau test like the Ljung -Box (due to small sample size) may be a more adequate test for autocorrelation. A computation of the Durbin-Watson h test (which accounts for the lagged dependent variable) shows that autocorrelation is still a problem in the regression. Again, the effect of adding the lagged dependent variable is that it reduces the number of observations by one.
Having estimated four different models (simple OLS, Cochrane-Orcutt of order one and two, OLS with lagged dependent variable), we found out that the best model which insulates against autocorrelation turns out to be the Cochrane-Orcutt of order 2. In that case proper inferencing may be done. We find that the AR(2) process illustrates well the phenomena in the change in consumption. Any shock will have a larger effect the following year as the shock has its total effect. However, after the second period, agents will adjust to the shock but rather sluggishly, hence showing the negative coefficient associated with it. Both lagged coefficients are statistically significant. All the other exogenous variables are statistically significant at the 5 % level. Only consumption and inflation rates tend to fail the test at the 1 % level. The results are also consistent in terms of macroeconomics. The inflation rate and the rate of interest are found to have a negative impact on the logarithm of consumption. Moreover the elasticity of consumption with respect to income is positive and high at 0.95. Overall, the regression is blessed with good goodness of fit with the explanatory variables explaining 99 % of the variation in the model. This is again confirmed by the low residual sum of squares. Taken altogether the regressors are found to be statistically significant.
In a nutshell, the above analysis has shown that the regression equation suffers from autocorrelation and inference from the ordinary least squares model is going to give us invalid conclusions. Hence we attempted to remove the autocorrelation by considering three procedures., namely the Cochrane-Orcutt with one lagged error term, Cochrane Orcutt with two lagged error terms and finally, adding a lagged dependent variable in the model. We found that the previously positive autocorrelation amongst the error terms was controlled only in the Cochrane-Orcutt model with 2 residuals. It is worth noting however that all three models improve on the conventional Ordinary Least Squares results. The statistics used to detect autocorrelation were the Durbin Watson d statistic in the first three models and the Durbin Watson h statistic in the last model.
However, we note that many issues other than autocorrelation are worth taking into account. Clearly, given the small sample size of the regression model, proper inference is restrictive and the tests for autocorrelation might not be so powerful. A proper remedy might be to take quarterly observations leading to a higher power by the autocorrelation tests. Moreover, it is worth mentioning that the conventional goodness of fit measures might be reduced once non-stationarity of the variables is taken into account.